## Necessary and Sufficient Conditions for Isomorphism

**Point of post:** This post will give a nice result which will give us necessary and sufficient conditions for a map between vector spaces to be an isomorphism. I will cite this post every now and then when I would like to shorten a proof.

**Theorem:** Let and be finite-dimensional vector spaces over the field . Then, a linear mapping is an isomorphism if and only if there exists bases for and for such that

is a bijection.

**Proof:** Suppose first that there existed bases and such that the above is true and assume without loss of generality that . To prove injectivity suppose that . Then, since is a basis for we know that and for some . Thus, by the linearity of we have that

But, this says that

and since is a basis for it follows that and so . Now, to prove surjectivity we merely note that for any we have that there is some such that . But, it follows that

from where surjectivity follows. Thus, is a bijective linear mapping and thus an isomorphism.

Conversely, it is trivial that if is an isomorphism that the image under any basis for is a basis for , and since the restriction of a bijection is a bijection the conclusion follows.

[…] linearity. By a previous post we may conclude that is an isomorphism. Also, we note that if and we see […]

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